Galois deformations are an important tool in Wiles' arsenal
for proving FLT. Are there any more elementary aspects (I'm
thinking of 1-dimensional Galois representations attached to
number fields) that would help the novice in better
understanding what's going on?

Here's what I have in mind. Let
$\rho: G_{\mathbb Q} \longrightarrow {\mathbb C}^\times$
be a 1-dimensional representation of the absolute Galois group
of the rationals factoring over some finite extension. Given a
Dirichlet character
$\chi: GL_1({\mathbb Z}/N{\mathbb Z}) \longrightarrow {\mathbb C}^\times$,
we can find representations
$\rho_\chi: Gal(K/{\mathbb Q}) \longrightarrow {\mathbb C}^\times$
for any cyclotomic extension $K = {\mathbb Q}(\zeta_N)$.
Call $\rho$ modular if there is a $\chi$ such that $\rho = \rho_\chi$.
The statement that every $\rho$ coming from an abelian extension is
modular is the theorem of Kronecker-Weber, and in this form it can be
proved using Galois deformations along the lines of Wiles' proof
(see Tunnell's proof in Kowalski's notes).
BTW if anyone knows a source for this result that is more readable
than Kowalski's notes (which I discovered just a couple of days
ago and haven't studied in detail yet) I'm all ears.

Question: Are there other similarly "elementary" questions, for
example in embedding problems or inverse Galois theory, that can
be described in terms of Galois deformations?

normally when we talk about Galois deformation, we talk about p-adic representations, right?
–
naturaMar 2 '10 at 1:19

2

In the 1-d case the universal deformation rings that show up are the ones usually called "Lambda" in Iwasawa theory (and this is trivial to check---you're just basically forming profinite group rings in this case). If I remember correctly Mazur makes some remarks about this in his original paper on the subject. By the way I think it's better to let rho be a p-adic Galois representation and say it's modular if it's a product of an integral power of the cyclo char by a finite order char. Then you get modular iff de Rham iff comes from an algebraic automorphic form, which is the analogy you want.
–
Kevin BuzzardMar 2 '10 at 12:37

2

Someone who writes out all the details of the proof of Kronecker-Weber using this approach will do a great service to the community (and to himself, in the bargain).
–
Chandan Singh DalawatMar 3 '10 at 8:06

2

"...Kronecker-Weber...". I'm not sure it's as easy as that. Wiles/Taylor use the Poitou-Tate exact sequence as an input into proving their R=T theorems and this exact sequence encodes most of global CFT already. So you might find the argument is circular. In fact if I remember correctly I think Larry Washington wrote an article in Cornell-Silverman-Stevens explaining all this. Nowadays one needs weaker numerical criteria to get R=T results but I'm still not at all convinced that you can get around CFT going in as an input.
–
Kevin BuzzardMar 3 '10 at 9:59

8

Dear Kevin and Brian, I've looked over the Tunnell notes before, and I don't think they are circular. Of course, they don't give the quickest proof of Kronecker--Weber by any means; but nor were they intended to. Rather, they were supposed to give a blueprint for the arguments of Wiles and Taylor in a simpler setting, where the deformation-theoretic arguments could be made by hand (hence, no cohomological theorems needed, and no circularity).
–
EmertonMar 3 '10 at 14:15

2 Answers
2

Late answer in continuation to #4: if I remember right, at first Tunnell thought he would need Poitou-Tate to complete the Kronecker-Weber proof in this manner (which would make things rather unsatisfactory, as people have said), but in the end -- for Kronecker-Weber only -- he saw that he only needed the local Kronecker-Weber theorem to complete the argument, which he then proved in the class (or maybe he just sketched the proof?).

Nigel Boston has a set of notes on the proof of FLT: http://www.math.wisc.edu/~boston/ (the notes is the link "Spring 2003 Math 869 Fermat's Last Theorem notes" under "Courses"). I haven't finished reading it, but it looks good.